According to a D.C. fire department report, ladder Truck 34 is ready to respond to an emergency, as a backup, in case other vehicles break down.

In reality, Truck 34 is in a scrap yard in Wisconsin, with no ladder, no back seat and no steering wheel.

It’s one of at least four trucks and six engine pumpers incorrectly listed in the department’s reserve fleet in a spreadsheet fire officials handed over last month to the D.C. Council’s public safety committee in advance of an oversight hearing.

The purpose was to show that the department is adequately managing its duties with existing equipment. The firefighters union says the discrepancies show the opposite. And while managers and union leaders agree the city sorely needs tens of millions of dollars’ worth of new trucks, the dispute highlights their disagreement about whether the need is affecting service.

The union uncovered the discrepancies by checking serial numbers, visiting Web sites where the city sells surplus vehicles and conducting a station-by-station and garage-by-garage inventory across the District.

When confronted with the information last week, the fire chief conceded the errors and announced that the deputy chief who had compiled the list would retire, ending his 28 years with the city.

It was the latest embarrassment for a fire department plagued by a string of mishaps and slow response times to emergencies. In recent weeks, an injured police officer was taken to the hospital by an ambulance from neighboring Prince George’s County because District vehicles were unavailable. An elderly stroke victim was transported in the back of a fire engine because the nearest ambulance was seven miles away. A man who had suffered a heart attack waited 29 minutes for an ambulance to arrive and later died.

The broken-down fire vehicles include Foam 1, a firefighting tanker assigned to be on hand for landings of the vice president’s helicopter at his residence in Northwest Washington. The truck has been out of commission for three months, prompting the department to seek help from the U.S. Navy to secure the landing site.

Council member Tommy Wells (D-Ward 6), who chairs the committee that received the faulty information, said he wants to know how many other times the department has issued bad numbers. He scheduled a hearing March 28 to press for answers.

“This undercuts the confidence I have in the information that was provided to me,” Wells said. “It means that oversight will be stepped up going forward. . . . I do not believe I have yet gotten a full explanation from the administration.”

The fire chief, Kenneth B. Ellerbe, said in a statement that the retiring deputy “used an old fleet schedule that had not properly excluded apparatus that had been removed from our inventory.”

The statement says that “as a result of his oversight, inaccurate information was reported” on a list that “included apparatus that were no longer in the department’s fleet.”

The deputy chief could not be reached for comment.

Dabney Hudson, the second vice president of firefighters union Local 734, said that misstating the department’s vehicle inventory could complicate replacement efforts, because it is not easy to push through truck purchases that can each cost $1.7 million and engines in the $700,000 range.

“We’re in a very bad situation,” said Hudson, a sergeant assigned to a truck company on Capitol Hill. “This is not something that’s an easy fix.”

The list provided by the fire department said the department has 29 truck companies, of which 16 are in service and the rest in reserve. The report also listed 65 engines, with 33 on the front line and the remainder in reserve. Hudson said at least two trucks on the reserve list were sold, including the one in Wisconsin, and two are being repaired. And at least six engines purportedly in reserve had been sold or are being repaired.

Lon Walls, the fire department’s spokesman, said that when Ellerbe became chief in 2010, he inherited an massive “apparatus deficit” of dozens of engines, trucks, rescue squads and ambulances. Since 2011, the department has established an “aggressive” plan to purchase nearly all the needed vehicles and even a boat.

But union officials say too many vehicles are in the shop or broken. A day after city officials announced that two reserve ambulances were available, a medic found one of them empty of all its equipment, including its stretcher.

One of the broken trucks is Foam 1, supposedly based at a station near L’Enfant Plaza. Built in 2001, it has been out of service since November.

Officials said the 2011-model Foam 2 truck has been pulling double duty — handling helicopter landings for the president at the White House and the vice president at his home at the Naval Observatory.

Only when both officials arrive simultaneously does a foam truck have to be dispatched from a U.S. Navy station in Southwest Washington.

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